Abstract In a previous paper we proposed a rigorous formalism for determining the electromagnetic field in grating coupler-waveguide systems. This formalism, developed for the case of TE polarization ( E parallel to the grooves of the grating), is here extended to the TM case ( H parallel to the grooves). Furthermore, we describe a systematic method for finding and characterizing the coupling resonances which appear when the system is illuminated by a plane wave. Some numerical results are given including the case in which there is some absorption by the photoresist.